a generalized "rotating" fluid equation model enabled by a fractional energy Hilbert space H(1/2)
which is about a well posed variation model of the NSE in a H(-1/2) framework. This goes along with a related energy measure of the "rotating" fluids which corresponds to the H(1/2) Hilbert space norm and a related classical (averaging/aggregating) energy measure which corresponds to the L(2)(H(1/2)) norm.
In the following we give some challenges and hurtles of current situation which lead to this solution concept alternatively to the "standard" H(0) framework.
In case of n=3 whether a strong solution of the non-linear, non-stationary Navier-Stokes equations exists on whole given time-interval without any smallness condition is a fundamental open mathematical problem. Answers exists, e.g. for initial value function with a non-smallness regularity assumption, which is basically about the 1/2 scale of Sobolev spaces as domain of the initial value functions (Fujija-Kato/Sohr).
We emphasis, that the pressure of the NSE requires neither initial value nor boundary conditions, but the NSE problem statement is about well posedness of the NSE system. The special role of the Hilbert space H(1/2) is related to the Laplace boundary layer problem, accompanied by J. Plemelj's extended Green formulas with respect to reduced regularity assumptions to the underlying domains.
Solutions of the divergence problem with homogenous Dirichlet data are based on Calderon-Zygmund theory or rely on the Stokes equation with inhomogenous data. Corresponding solutions with Sobolev space domains are built on the Bogovskii solution operator, which can be extended continously to an operator acting from W(s,p) to W(s+1,p) for s>-2+1/p (Geissert M., Heck H., Hieber M.)
The questions concerning the existence of weak solutions of the non-linear, non-stationary Navier-Stokes equations have been basically answered. Corresponding "extrapolation" to related strong solutions by density arguments are different per problem category (linear/non-linear, stationary/non-stationary, space dimension), basically due to the structure of the Stokes operator (and its related global/local "incompatibility" between pressure and velocity), the Serrin gap (related to the non-linear term) according to Sobolev embedding theorems and the logarithm convexity of the Sobolev spaces.
Regularity and uniqueness of weak solutions for n=3 are still pending (Giga/Sohr), basically due to the Serrin gap challenge, which is a consequence of the Sobolev embedding theorem. An existence and uniqueness result of solutions in L(r) of the Navier-Stokes initial value problem is given in Giga/Miyakawa (as well, that the solutions are smooth up to the boundary if the external force is smooth). The essential step in Giga/Miyakawa is an estimate of the nonlinear term (u,grad)u, based on the resolvent of fractional powers of the Stokes operator, applying the calculus of pseudodifferential operators (Giga Y., "Weak and strong solutions of the Navier-Stokes initial value problem", Lemma 3.2). We emphasis that the fractional powers of the Stokes operator generates corresponding Hilbert scale, which enables e.g. (weak) a variational representation of the NSE initial value problem with respect to negative Hilbert scale inner products (e.g. the inner product of the Hilbert space H(-1/2)). The same is supported by lemma 2.5.2 (Sohr H., p. 152), which is about the boundedness (with corresponding appropriate definition of domains) of negative fractional Stokes operators applied to the Helmholtz projection to define extended, bounded Helmholtz projection operators with respect to related (negative) Hilbert space inner products (Sohr H., p. 264 ff.)
Basically all existence proofs of weak solutions of the Navier-Stokes equations are given as limit (in the corresponding weak topology) of existing approximation solutions built on finite dimensional approximation spaces. The approximations are basically built by the Galerkin-Ritz method, whereby the approximation spaces are e.g. built on eigenfunctions of the Stokes operator or generalized Fourier series approximations. Thereby the quality of the approximations spaces itself seems to be not relevant, e.g. special "quasi-optimal" approximations properties of finite element approximation spaces weren’t applied for those solutions.
The "power 3" challenge with its corresponding (time variable related) blow-up effect is a given due to the Sobolev embedding theorem and corresponding Serrin scale conditions. This is basically a consequence of an underlying ordinary differential (Riccati) problem. A reduced Hilbert scale factor of the (energy) Hilbert space is proposed to reduce the problematic "power of 3" term down to "power of 2" (still a challenging Riccati ODE) term, as one element of a solution concept. The consequences to the physical model parameters would be to define a pressure force based on a non-harmonic pressure function at the boundary layers.
From a Galerkin approximation method perspective addressing the non-regular behavior of the N-S-E solution(s) at zero and blow-up time go in line with local convergence analysis for non-linear parabolic equations with reduced regularity of the initial value function and/or not fulfilled compatibility conditions. The transformation of the (one dimensional) free boundary Stefan problem to a (non-linear) fixed boundary value equation provides the simplest model problem for such situations, which still has open questions, as well.
From a functional analysis perspective the above goes in line with (hyper) singular integral operators, which correspond to single / double layer potential operator, the tangential derivative ofthe single layer operator and the normal derivative of the double layer operator.
The idea: an appropriately defined distributional Hilbert space framework
In an appropriately defined distributional Hilbert space framework the incompressible (!) N-S-E are well posed. The shift on the Hilbert scale to the left “closes” the Serrin gap in case of n=3 and"time-weighted" norms "control" the singular velocity and pressure behavior for t --> 0. At the same time the Hilbert scale shift jeopardies the application of the Sobolev embedding theorem to make conclusions about classical solutions. By standard functional analysis this should enable the building of a counterexample (following the idea of J. Heywood) that existing weak solutions in "standard" scaled Sobolev spaces lead to "strong solution".
Just from a common sense feeling the same argument should not be valid in case for compressible "fluids". This would be in line with J. Plemelj's concept of a mass element, replacing an only mass density, which requires less regularity assumptions to enable the Hilbert scale shift to the left, while keeping the option to apply the Sobolev embedding theorem. This would also be in line with
[ArA] Arthurs A. M., "Complementary variational principles", where one example (§ 4.8) is about nonlinear variation method applied to compressible fluid flow.
Remark: It's suggested to apply J. Plemelj's theory also to the still not solved "radiation problem" (R. Courant, D. Hilbert, "Methods of mathematical physics, II", 1937, VI, §5, section 6, VI §10, section 3).
J. Plemelj proposed an alternative definition of a normal derivative, based on Stieltjes integral. It requires less regularity assumptions than standard definition; the "achieved" "regularity reduction"is in the same size as a reduction from a C(1) to a C(0) regularity. Plemelj's concept is proposed to be applied to ensure physical model requirements modeled by normal derivatives within a distributional Hilbert space framework.
The concept of Pseudo-Differential operators is about the study of the differential and integral operators within the same algebra of operators. The concept of Hilbert scale is about an appropriate Hilbert space with respect to the eigenpairs of self-adjoint, positive definite operators with corresponding order (respectively coercive operators in combination with the Garding inequality). The study of uniqueness and existence of weak solutions of the Navier-Stokes equation is about appropriate weak variation representation of the NSE and related approximation solutions, which converge to the NSE solution(s).
For the quasi-optimal approximation estimates of Ritz-Galerkin method in Hilbert scales (which, of course, also includes collocation methods) we refer to
[BrK1] Braun K., "Interior Error Estimates of the Ritz methods for Pseudo-Differential Equations".
Defining the right Hilbert space framework goes along with defining the proper Pseudo-Differential operators with their related domains (!). In this context we refer to the model (singular integral) Pseudo-Differential operators in
[BrK2] Braun K., "An alternative quantizationof H=xp"
[BrK3] Braun K., "A new ground state energymodel"
[LiI] Lifanov, I. K., Nenashev A. S., "Generalized functions on Hilbert spaces, singular integral equations, and problems of aerodynamics and electrodynamics".
For the linkage to the normal derivative concept of Plemelj (especially to the double layer potential, as well as its normal derivative, a hyper singular operator of Calderon-Zygmund type, which is essentially a once differentiation operator) we refer to
[AmS] Amini S., "On Boundary Integral Operators for the Laplace and the Helmholtz Equations and Their Discretization":
- the operator defined by the normal derivative of the single layer potential is the dual operator of the double layer potential
- the operator defined by the tangential derivative of the single layer potential is the Hilbert transform (S(0)). It is discontinuous on the surfaces with a jump according to the Plemelj formula
- the operator of the normal derivative of the double layer potential is a hyper-singular operator, which behaves as the derivative of the Hilbert transform operator (S(1)). It is continuous (!) on the surface.
The terms "vortex density" and "rotation of a fluid element" are sometimes used synonymy. The rotation of a vector field v describes the vortex field of v. The corresponding vortex force at a point x is defined as the product of the constant normal vector of all (infinitesimal small) areas F containing x with the rot(v). The vortex density in combination with the concept of"circulation" (L. Prandtl) is interpreted as the root cause of the "rotation" of a fluid element. The definition of"circulation" is integral part of the definition of "vortex force". We propose the use Plemelj's concept/definition of a mass element (alternatively to a mass density) to define a "vortex" of a fluid element, which would require less regularity assumptions to the vector field v than H(1), but being still consistent with the definitions of fluid density, vortex force and rotation in case of corresponding H(1)-regularity assumptions.
We note that
- moving from the "convection form" to the"rotation form" goes along with a movement from "kinematic pressure" to "Bernoulli pressure"
- the Euler equation is nonlocal, i.e. one cannot compute the time derivative of the solution u at (x,t) only from the knowledge of the function u in the neighborhood of x at the time t. This gets proven by taking the divergence of the NSE, which gives the Laplacian of the pressure p. Therefore the pressure p is determined by local information, but not the gradient of p (P. Constantin, "On the Euler equations of incompressible fluids").
References of the following three selected J. A. Nitsche papers
[NiJ1] Nitsche J. A., "L(infinitety) boundedness of the FE Galerkin operator for parabolic problems"
[NiJ2] Nitsche J. A., several papers in the context of "Finite element approximations and non-linear parabolic differential equation"
[NiJ3] Nitsche J. A., "Direct proofs of some unusual shift theorems".
ad [NiJ1] : The paper gives an ("optimal", i.e. shift on Hilbert scale by the factor"2") shift theorem for the heat equation based on Fourier transforms estimates. The "trick" (which is about changing the order of integration with respect to the time-variable) to prove an optimal Sobolev scale shift by scale factor "-2" (analogue to the elliptic Laplace equation) is proposed to be applied for appropriately defined time-depending norm estimates for the Oseen kernel in the context of harmonic analysis for solving the incompressible Navier-Stokes equations.
The link of [NiJ1], [NiJ2] to the N-S-E is given by the Leray-Hopf operator and its Oseen kernels. The Fourier transformations of the Oseen kernel (see Lerner N., "A note on the Oseen kernels" ) are required to apply the "heat equation shift theorem trick", in [NiJ1].
ad [NiJ3] : For the Stokes equation an unusual shift theorem (negative norm estimates) has been proven by solving auxiliary problems, building on the Cauchy-Riemann differential equations. The effect is that the Stokes problem is decoupled into two elliptic problems. The definition of the auxiliary functions (w,z) for given C-R- relationships for (u,v) can be interpreted as defining curl(u) and div(v) as representations of the C-R equations (n=2). For n=3 the curl operator (which is for n=2 one of the two C-R equations) is linked to the Leray-Hopf projector P (and therelated Riesz operators) by the following properties (M. Lerner):
- the commutator (P,curl) vanishes
- P(curl) = curl
- (curl(u),curl(v)) =(gradP(u),gradP(v)) = D(P(u),P(v)).
We note that curl(u) describes the mean rotation of a fluid and 1/2*rot(u) gives the mean value of the angular velocity.
Replacing the gradient operator by the Calderon-Zygmund operator enables the definition of a "curl"-operator with distributional Hilbert space domain. This follows the same idea concerning alternative "energy inner product in distributional Hilbert spaces" ([BrK1]) to overcome current issues with different domain regularity/scope of momentum and location operator.
The conjecture is that with the above shift theorems for appropriately defined norms can be proven for both, velocity and pressure, anticipating the divergent behavior for t --> 0.
We emphasis that the L(2)-norm of the (Oseen) tensor kernel of the solutions of the incompressible N-S-E can be estimated by t*exp(-a) with a=1/4 ([CaM] ), which is the same (space dimension n independent!) divergence behavior as for the one-dimensional Stefan problem with non-regular initial value function [NiJ4] .
As a consequence for the space dimension n=3 the „natural" energy norm H(1) is being replaced by an appropriate alternative with reduced regularity requirements. The concept of J. Plemelj still keeps the physical model relevance and, at the same time, strengths the capabilities of a potential. For the non-stationary case this norm is modified by timely-weight (integral-dt) norms. For such a Hilbert space there is a quasi-optimal shift theorem for the NSE. The initial value function for the pressure is basically of Dirac function type. We note, that the embedding properties of the Dirac function into the continuous function space are depending from the space dimension. The same is valid for the Serrin gap. The correspondingly required reduced regularity assumptions to the normal derivative are "delivered" by the concept of J. Plemelj ([PlJ]), respectively the corresponding Pseudo-Differential operator.
"Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas"